# KL divergence between sample and true (multivariate normal) distribution

I was wondering, whether there is a possible interpretation of the KL-Divergence between sample and true distribution in terms of probabilities. E.g. given $$P=\mathcal{N}\left(\mu,\Sigma\right)$$ and $$Q=\mathcal{N}\left(\hat{\mu},\hat{\Sigma}\right)$$ we have $$$$D_{KL}\left(P||Q\right)=\frac{1}{2}\left[\log\frac{|\hat{\Sigma}|}{|\Sigma|} - d + \text{tr} \{ \hat{\Sigma}^{-1}\Sigma \} + (\hat{\mu} - \mu)^T \hat{\Sigma}^{-1}(\hat{\mu} - \mu)\right]$$$$ My question refers to the fact that the (scaled) Mahalanobis Distance (which is part of the above) $$$$\frac{T(T-N)}{(T-1)N}(\hat{\mu} - \mu)^T \hat{\Sigma}^{-1}(\hat{\mu} - \mu)$$$$ has an $$F$$-distribution with $$(N,T)$$ degrees of freedom (where $$N=$$ dimension, $$T=$$ sample size) and therefore yields a probabilistic "value" to judge how "far" $$\hat{\mu}$$ is away from $$\mu$$.